On groups presented by inverse-closed finite convergent length-reducing rewriting systems

We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of $\mathbb Z$) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite convergent length-reducing rewriting system is not plain is in $\mathsf{NP}$. A "yes" answer would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite convergent length-reducing rewriting systems is in $\mathsf{PSPACE}$.